1. If an out-of-command unit is Formation Broken, can it reform rather than move (if allowed by Orders) on the basis that this will allow it to move back into command on a subsequent Activation?
See the last bullet under 5.4 in the series rules. The unit can reform only via the Army Commander.
2. If a two-hex HI unit is performing Rout movement and there is a one-hex gap, can it move through the gap sideways? This is what would happen in real life, but it would mean that one half of the unit would probably not expend its full movement allowance.
Dunno. I'm surprised the rules don't specify. I would say that the unit moves through the gap. It's not really even "sideways," since at that point the unit is basically a panicked mob - the facing of the chit doesn't reflect any kind of actual order in the unit. You should ask this on CSW if Ben doesn't pop in here.